Optimal. Leaf size=81 \[ \frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.0383484, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2240, 2234, 2204} \[ \frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2240
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x+c x^2} x \, dx &=\frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{b \int f^{a+b x+c x^2} \, dx}{2 c}\\ &=\frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\left (b f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c}\\ &=\frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{b f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.0644649, size = 81, normalized size = 1. \[ \frac{f^{a+b x+c x^2}}{2 c \log (f)}-\frac{\sqrt{\pi } b f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 79, normalized size = 1. \begin{align*}{\frac{{f}^{c{x}^{2}}{f}^{bx}{f}^{a}}{2\,c\ln \left ( f \right ) }}+{\frac{b\sqrt{\pi }{f}^{a}}{4\,c}{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12411, size = 144, normalized size = 1.78 \begin{align*} -\frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{3}{2}}} - \frac{2 \, c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{3}{2}}}\right )} f^{a - \frac{b^{2}}{4 \, c}}}{4 \, \sqrt{c \log \left (f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53768, size = 184, normalized size = 2.27 \begin{align*} \frac{2 \, c f^{c x^{2} + b x + a} + \frac{\sqrt{\pi } \sqrt{-c \log \left (f\right )} b \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}}}{4 \, c^{2} \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x + c x^{2}} x\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19864, size = 108, normalized size = 1.33 \begin{align*} \frac{\frac{\sqrt{\pi } b \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )}} + \frac{2 \, e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right )\right )}}{\log \left (f\right )}}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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